1. Field of the Invention
The invention is related to the field of electrical resistivity logging of oil and gas wells. More specifically, the invention is related to methods for improving the response of resistivity logs by correcting measurements for the effects of irregularities in the wall surface of the wellbore, and for drift in measuring circuits in the well logging instrument.
2. Description of the Related Art
Electrical resistivity well logging is used to determine the electrical resistivity of earth formations from within a wellbore drilled through the earth formations. Electrical resistivity well logging instruments and methods known in the art include so-called xe2x80x9cgalvanicxe2x80x9d resistivity instruments which impart electrical current to the earth formation in a predetermined flow pattern and measure voltage drop of the electrical current and the current magnitude to determine the resistivity of the earth formations. At any particular magnitude of current, the voltage drop is related to the resistivity of the formations surrounding the wellbore.
One type of galvanic resistivity instrument is the laterolog. The laterolog instrument uses focusing currents to constrain the path of the current used to measure voltage drop. By constraining the path of the measure current, the voltage drop in the measuring current can be more directly related to particular layers of the earth formations adjacent to the wellbore and the logging instrument. An example of a laterolog instrument is described in U.S. Pat. No. 5,585,727 issued to Fanini et al. Laterolog instruments work best where the resistivity of fluid in the wellbore is very low.
Another type of galvanic resistivity well logging instrument intended to overcome certain limitations of laterolog instruments is known as a xe2x80x9cmultiple electrode resistivity toolxe2x80x9d. A service using this type of galvanic well logging instrument is sold under the trade name xe2x80x9cHDLLxe2x80x9d by the assignee of this invention. See G. Itskovich et al, High-Definition Lateral Log Resistivity Device: Basic Physics and Resolution, 39th Annual Symposium, Society of Professional Well Log Analysts (1998). The multiple electrode resistivity tool includes a measuring current source connected to an electrode on an insulating sonde mandrel and a series of voltage measurement electrodes located at spaced apart positions from the source-connected electrode along the insulating mandrel. Electrical current is emitted into the wellbore from the source-connected electrode, is returned at the earth""s surface, or other physically distant location, and the voltage is measured at each of the measuring electrodes. Difference in potential is also measured between pairs of adjacent ones of the measuring electrodes. The voltage measured at each one of the electrodes, and the potential differences measured between the pairs of adjacent electrodes are used to determine the resistivity of the earth formations in both the uninvaded and invaded zones as well as the radial extent of the invaded zone by a process known in the art as xe2x80x9cinversionxe2x80x9d. This process is described in the Itskovich et al reference, supra.
A limitation to the multiple electrode resistivity tool is that the voltage difference measurements can be subject to substantial error if the wall of the wellbore is irregular, particularly when the earth formation resistivity is very large relative to the resistivity of the fluid in the wellbore. If the wall of the wellbore is irregular, the measuring current path can become distorted and the relationship between voltage difference and earth formation resistivity can be changed as a result.
Another method for determining the resistivity of earth formations is to cause electrical current to flow mainly along the wellbore and to determine the amount of the current which xe2x80x9cleaksxe2x80x9d laterally away from the wellbore into the earth formation. The amount of current leakage is related to the conductivity (inverse of resistivity) of the earth formation. This principle has been used to determine the resistivity of earth formations from inside a conductive steel pipe or xe2x80x9ccasingxe2x80x9d. See U.S. Pat. No. 4,748,415 issued to Vail, for example. The theory of this measurement can be explained as follows. Consider, for example, a xe2x80x9ccurrent tubexe2x80x9d in a wellbore having an axial dimension xcex94 along the wellbore wall adjacent to an earth formation interval of interest. The amount of lateral current leakage Ir into the earth formations is equal to the difference in current between the upper and lower axial limits of the formation interval of interest, this difference represented by the expression:
Ir=Izxe2x88x92xe2x88x92Iz+xe2x80x83xe2x80x83(1) 
where Izxe2x88x92 and Iz+ represent, respectively, the current flowing axially at the lower and upper limits of the earth formation interval of interest. The difference in currents may be expressed by a second difference of potentials, by an expression such as:                                           I            z            -                    -                      I            z            +                          =                                            π              ⁢                              xe2x80x83                            ⁢                              b                2                                                    R              m                                ⁢                                                    ∂                2                            ⁢              V                                      ∂                              z                2                                              ⁢          Δ                                    (        2        )            
In equation (2), Rm represents the resistivity of fluid (xe2x80x9cdrilling mudxe2x80x9d) in the wellbore and b represents the radius of the wellbore. The amount of current leakage is directly proportional to the voltage V at the wellbore wall, the inverse resistance of the current tube per unit length, Gt, and the axial dimension xcex94. This can be expressed as:
Ir=Vxc3x97Gtxc3x97xcex94xe2x80x83xe2x80x83(3) 
The inverse resistance of the current tube per unit length, Gt has physical dimension of (ohm-m)xe2x88x921. By combining equations (1) and (3), a well known xe2x80x9ctransmission linexe2x80x9d equation can be obtained for the potential V:                                                                         ∂                2                            ⁢              V                                      ∂                              z                2                                              =                                    α              2                        ⁢            V                          ⁢                  
                ⁢                              α            2                    =                                                    R                m                            ⁢                              G                t                                                    π              ⁢                              xe2x80x83                            ⁢                              b                2                                                                        (        4        )            
The second potential derivative directly follows the formation resistivity, reflected in factor Gt. This principle has been used, as described in the Vail ""415 patent, for determining resistivity of an earth formation from inside a conductive steel casing. The principle is particularly applicable to measurement of formation resistivity from inside a conductive casing where the contrast in resistivity between the wellbore and earth formation is very high. Where the wellbore includes a conductive casing, this contrast can exceed a factor of 109. It has not been determined that the principle described in the Vail ""415 patent has any application where such a large resistivity contrast between the wellbore and formation does not exist, as in the case where the wellbore does not include a conductive steel casing.
The invention is a method for determining the resistivity of earth formations penetrated by a wellbore. Electrical current is imparted to the wellbore and to the earth formations from a first electrode which is located on an insulating sonde mandrel disposed in the wellbore. The electrical current is returned at a second electrode also disposed on the sonde mandrel, at an axially spaced apart location from the first electrode. Voltage differences are measured between a first pair and a second pair of electrodes located on the mandrel between the first and the second electrodes. Circuits which are used to measure the voltage differences between the pairs of electrodes are then adjusted, until the measured voltage difference between the second pair of electrodes is substantially the same as the measured voltage difference between the first pair of electrodes. Then the electrical current is imparted from the first electrode, and is returned to an electrode located at the earth""s surface or other physically distant location. Measuring the voltage differences between the pairs of electrodes is repeated. A difference of the voltage differences is then determined. The difference of voltage difference is directly related to current leakage into the earth formation, and this current leakage determination is substantially unaffected by irregularities in the wall of the wellbore or drift in the circuits used to measure the voltage differences. The current leakage is directly related to the conductivity of the earth formations. The current leakage can be inverted to obtain resistivity of the earth formations.